0
$\begingroup$

My apologies if this has been asked before. I wasn't able to find an explanation that made sense to me on Google.

Suppose we have the equation below, which is written in Einstein notation:

$$y_{j} = x_iz_ix_j$$

and say that both $j$ and $i$ take on values of $1$ and $2$. Is this notation to be interpreted as

$$y_1 = (x_1z_1+x_2z_2)x_1$$ $$y_2 = (x_1z_1+x_2z_2)x_2$$

or as $$y_1+y_2 = (x_1z_1+x_2z_2)(x_1+x_2)?$$

In other words, does the index across the equals sign imply multiple equations, or adding up terms on each side of a single equation?

Context: I am an undergrad math major studying fluid flow. The conservation of momentum equation was written in Einstein Notation and I am having trouble understanding the meaning.

$\endgroup$
  • $\begingroup$ It's the first. Think about the simpler case $x_i = y_i$. This should mean that the vectors $x$ and $y$ are equal, not that their components sum to the same number. $\endgroup$ – knzhou Feb 21 '17 at 0:28
1
$\begingroup$

The correct interpretation is the first one: $$ y_j=x_iz_ix_j$$ Means that to obtain the jth component of $y$ you have to sum over the repeated indeces, i.e.

$$ y_j=\sum_{i=1}^2 x_iz_ix_j$$

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ Is it a coincidence that the two interpretations yield the same result in this case, or does it happen that way every time? I guess all we know is true is that the first implies the second. $\endgroup$ – EternusVia Feb 21 '17 at 0:28
  • $\begingroup$ It's coincidence. You can think at the definition of cross product using the Einstein notation as an example. $\endgroup$ – Alessandro Zunino Feb 21 '17 at 0:34

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.